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<b>consider </b><font color="Maroon" title="c1">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:21_1"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E3:21_1"/>
<font color="Maroon" title="c1">f</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:21_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<a NAME="E2:21_1_1"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:21_1_1"/><b>then </b>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c2">i</font>) is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E1:21"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <b>by </b><i><a class="txt" href="pre_circ.html#E2:21_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E2:21_1_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">f</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<a NAME="E5:21_1"/>
<font color="Maroon" title="c2">f</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:21_1_2"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E2:21_1_2"/><b>then </b>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c3">i</font>) is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E1:21"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M6</a>(<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>)
 <b>by </b><i><a class="txt" href="pre_circ.html#E2:21_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E1:21_1_2"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">f</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<b>thus </b><a NAME="E7:21_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>by </b><i><a class="txt" href="pre_circ.html#E2:21_1"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>


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